Francisco Bernal

Michael B. Giles, Francisco Bernal

This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexity to achieve an $\varepsilon$ root-mean-square error is $O(\varepsilon^{-2}\, |\!\log \varepsilon|^3)$.

René Aïd, Francisco Bernal, Mohamed Mnif et al.

This work presents a novel policy iteration algorithm to tackle nonzero-sum stochastic impulse games arising naturally in many applications. Despite the obvious impact of solving such problems, there are no suitable numerical methods available, to the best of our knowledge. Our method relies on the recently introduced characterization of the value functions and Nash equilibrium via a system of quasi-variational inequalities. While our algorithm is heuristic and we do not provide a convergence analysis, numerical tests show that it performs convincingly in a wide range of situations, including the only analytically solvable example available in the literature at the time of writing.

Francisco Bernal, Juan A. Acebrón

We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates--resulting in an overall acceleration of PDD for a given error tolerance. The key ingredient of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement of between one and two orders of magnitude over the previous version of PDD.

Francisco Bernal

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa's method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau's problem, Hele-Shaw flow and the Monge-Ampère equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians.

Francisco Bernal, Gonçalo dos Reis, Greig Smith

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.

Francisco Bernal

Despite its generality and powerful convergence properties, Milstein's method for functionals of spatially bounded stochastic differential equations is widely regarded as difficult to implement. This has likely prevented it from being utilised in applications. In this paper, we design and analyse in detail one such implementation. The presented method turns out to be on par with other, popular schemes in terms of computational cost---but with a (nearly) linear weak convergence rate under the usual smoothness requirements on coefficients and boundary. Two byproducts of theoretical interest are a new, non-standard rank-one update formula, and a connection between numerics of bounded diffusions and Eikonal equations. Three examples are worked out, confirming the accuracy and robustness of the method.

Francisco Bernal, Gail Gutiérrez

A general and easy-to-code numerical method based on radial basis functions (RBFs) collocation is proposed for the solution of delay differential equations (DDEs). It relies on the interpolation properties of infinitely smooth RBFs, which allow for a large accuracy over a scattered and relatively small discretization support. Hardy's multiquadric is chosen as RBF and combined with the Residual Subsampling Algorithm of Driscoll and Heryudono for support adaptivity. The performance of the method is very satisfactory, as demonstrated over a cross-section of benchmark DDEs, and by comparison with existing general-purpose and specialized numerical schemes for DDEs.

Francisco Bernal, Juan A. Acebrón

We review, implement, and compare numerical integration schemes for spatially bounded diffusions stopped at the boundary which possess a convergence rate of the discretization error with respect to the timestep $h$ higher than ${\cal O}(\sqrt{h})$. We address specific implementation issues of the most general-purpose of such schemes. They have been coded into a single Matlab program and compared, according to their accuracy and computational cost, on a wide range of problems in up to ${\mathbb R}^{48}$. The paper is self-contained and the code will be made freely downloadable.